Proof approach ! ? 14 L ( W 1 ) , L ( W 2 ) reg. sep L ( W 1 ) ∩ L ( W 2 ) = L ( W 1 × W 2 ) = � W 1 × W 2 has inductive invariant
Proof approach ! ? 14 L ( W 1 ) , L ( W 2 ) reg. sep L ( W 1 ) ∩ L ( W 2 ) = L ( W 1 × W 2 ) = � W 1 × W 2 has inductive invariant
Proof approach ! ? 14 L ( W 1 ) , L ( W 2 ) reg. sep L ( W 1 ) ∩ L ( W 2 ) = L ( W 1 × W 2 ) = � W 1 × W 2 has inductive invariant
Finitely represented invariants The desired implication does not hold. 15 Call an invariant X finitely represented if X = Q ↓ for Q finite
Finitely represented invariants The desired implication does not hold. Recall: iff upward-closed sets have finitely many minimal elements. No such statement for downward-closed sets and maximal elements! 15 Call an invariant X finitely represented if X = Q ↓ for Q finite ( S , ⩽ ) well quasi order (wqo)
Finitely represented invariants The desired implication does not hold. We can show: Theorem 15 Call an invariant X finitely represented if X = Q ↓ for Q finite Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable.
Proof approach II ! 16 L ( W 1 ) , L ( W 2 ) reg. sep L ( W 1 ) ∩ L ( W 2 ) = L ( W 1 × W 2 ) = � ✓ ✗ W 1 × W 2 has fin.-rep. invariant
Proof approach II ! 16 L ( W 1 ) , L ( W 2 ) reg. sep L ( W 1 ) ∩ L ( W 2 ) = L ( W 1 × W 2 ) = � ✓ ✗ W 1 × W 2 has fin.-rep. invariant
Ideals Finitely represented invariants do not necessarily exist. Solution: Ideals Definition Lemma 17 For WSTS W , let � W be its ideal completion. [KP92][BFM14,FG12] L ( W ) = L ( � W ) .
Ideals Finitely represented invariants do not necessarily exist. Solution: Ideals Definition Lemma Proposition 17 For WSTS W , let � W be its ideal completion. [KP92][BFM14,FG12] L ( W ) = L ( � W ) . If X is an inductive invariant for W , then its ideal decomposition Idec ( X ) ↓ is a finitely-represented inductive invariant for � W .
Proof Putting everything together: This finitely-represented invariant gives rise to a regular separator. 18 If W 1 , W 2 are disjoint, W 1 × W 2 admits an invariant X . Then Idec ( X ) ↓ is a finitely-represented invariant for W 1 × W 2 ∼ � = � W 1 × � W 2 .
Proof Putting everything together: If two WSTS languages are disjoint, Theorem We have shown: separator. This finitely-represented invariant gives rise to a regular 18 If W 1 , W 2 are disjoint, W 1 × W 2 admits an invariant X . Then Idec ( X ) ↓ is a finitely-represented invariant for W 1 × W 2 ∼ � = � W 1 × � W 2 . one of them finitely branching or deterministic or ω 2 , then they are regularly separable.
Proof details: From fin.-rep. invariants to regular separators
From invariants to separability Theorem 19 Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable.
From invariants to separability Theorem 19 Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable. Assume Q ↓ is invariant. Idea: Construct separating NFA with Q as states
From invariants to separability s s F 1 s Q s s Q F initial for some c c c c Theorem Q s s Q I Definition 19 Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable. A = ( Q , → , Q I , Q F ) where
From invariants to separability Theorem Definition Q F s s Q s F 1 19 Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable. A = ( Q , → , Q I , Q F ) where Q I = { ( s , s ′ ) ∈ Q | ( c , c ′ ) ⩽ ( s , s ′ ) for some ( c , c ′ ) initial }
From invariants to separability Theorem Definition 19 Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable. A = ( Q , → , Q I , Q F ) where Q I = { ( s , s ′ ) ∈ Q | ( c , c ′ ) ⩽ ( s , s ′ ) for some ( c , c ′ ) initial } Q F = { ( s , s ′ ) ∈ Q | s ∈ F 1 }
From invariants to separability Theorem a � a 19 Definition Let W 1 , W 2 WSTS, W 2 deterministic. If W 1 × W 2 admits a finitely-represented inductive invariant, then L ( W 1 ) and L ( W 2 ) are regularly separable. A = ( Q , → , Q I , Q F ) where Q I = { ( s , s ′ ) ∈ Q | ( c , c ′ ) ⩽ ( s , s ′ ) for some ( c , c ′ ) initial } Q F = { ( s , s ′ ) ∈ Q | s ∈ F 1 } ( r , r ′ ) ∈ Q in A ⩽ � ( t , t ′ ) ∈ S 1 × S 2 Q ∋ ( s , s ′ ) in W 1 ×W 2
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
20 a the configurations from Q . c b a c b Behavior of A q 1 ↓ q 3 ↓ q 0 ↓ • • F 1 × S 2 • • q 2 ↓ • • • • A over-approximates the behavior of the product system using
Proving separability: Inclusion Lemma 21 L ( W 1 ) ⊆ L ( A ) .
Proving separability: Inclusion Lemma Proof. 21 L ( W 1 ) ⊆ L ( A ) . − → d of W 1 Any run c w synchronizes with the run of W 2 for w in the run ( c , c ′ ) w → ( d , d ′ ) of W 1 × W 2 . −
Proving separability: Inclusion Lemma Proof. 21 L ( W 1 ) ⊆ L ( A ) . − → d of W 1 Any run c w synchronizes with the run of W 2 for w in the run ( c , c ′ ) w → ( d , d ′ ) of W 1 × W 2 . − This run can be over-approximated in A .
Proving separability: Inclusion Lemma Proof. 21 L ( W 1 ) ⊆ L ( A ) . − → d of W 1 Any run c w synchronizes with the run of W 2 for w in the run ( c , c ′ ) w → ( d , d ′ ) of W 1 × W 2 . − This run can be over-approximated in A . If d is final in W 1 , the over-approximation of ( d , d ′ ) is final in A .
Proving separability: Disjointness Lemma 22 L ( W 2 ) ∩ L ( A ) = � .
Proving separability: Disjointness Lemma Proof. 22 L ( W 2 ) ∩ L ( A ) = � . Any run of A for w over-approximates in the second component the unique run of W 2 for w .
Proving separability: Disjointness Lemma Proof. 22 L ( W 2 ) ∩ L ( A ) = � . Any run of A for w over-approximates in the second component the unique run of W 2 for w . If w ∈ L ( W 2 ) ∩ L ( A ) then some run of A reaches a state ( q , q ′ ) with - q final in W 1 (def. of Q I ) - q ′ final in W 2 ( w ∈ L ( W 2 ) + argument above)
Proving separability: Disjointness Lemma Proof. 22 L ( W 2 ) ∩ L ( A ) = � . Any run of A for w over-approximates in the second component the unique run of W 2 for w . If w ∈ L ( W 2 ) ∩ L ( A ) then some run of A reaches a state ( q , q ′ ) with - q final in W 1 (def. of Q I ) - q ′ final in W 2 ( w ∈ L ( W 2 ) + argument above) Contradiction to F 1 × F 2 ∩ Q ↓ = � !
Proof details: The ideal completion and fin.-rep. invariants
Finitely represented invariants Lemma A similar result for downward-closed subsets and maximal elements does not hold. 23 Let U ⊆ S be an upward-closed set in a wqo. There is a finite set U min such that U = U min ↑ .
Finitely represented invariants Lemma A similar result for downward-closed subsets and maximal elements does not hold. Example: 23 Let U ⊆ S be an upward-closed set in a wqo. There is a finite set U min such that U = U min ↑ . Consider N in ( N , ⩽ ) Intuitively, N = ω ↓
Finitely represented invariants Lemma A similar result for downward-closed subsets and maximal elements does not hold. Consequence: Finitely represented invariants may not exist! Solution: Move to a language-equivalent system for which they always exist. 23 Let U ⊆ S be an upward-closed set in a wqo. There is a finite set U min such that U = U min ↑ .
• directed: Ideals • non-empty • downward-closed x y z x z y z 24 Let ( S , ⩽ ) be a wqo An ideal I ⊆ S is a set that is
Ideals • non-empty • downward-closed 24 Let ( S , ⩽ ) be a wqo An ideal I ⊆ S is a set that is • directed: ∀ x , y ∈ I ∃ z ∈ I : x ⩽ z , y ⩽ z
Ideals • non-empty • downward-closed Example 1: 24 Let ( S , ⩽ ) be a wqo An ideal I ⊆ S is a set that is • directed: ∀ x , y ∈ I ∃ z ∈ I : x ⩽ z , y ⩽ z For each c ∈ S , c ↓ is an ideal
Ideals • non-empty • downward-closed Example 2: 24 Let ( S , ⩽ ) be a wqo An ideal I ⊆ S is a set that is • directed: ∀ x , y ∈ I ∃ z ∈ I : x ⩽ z , y ⩽ z Consider ( N k , ⩽ ) The ideals are the sets u ↓ for u ∈ ( N ∪ { ω } ) k
Ideal decomposition Lemma ([KP92]) inclusion-maximal ideals in D 25 Let ( S , ⩽ ) be a wqo For D ⊆ S downward closed, let Idec ( D ) be the set of Idec ( D ) is unique, finite and we have ∪ D = Idec ( D )
Ideal completion Definition ([BFM14,FG12]) F F T defined by Post a Idec Post a 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with
Ideal completion Definition ([BFM14,FG12]) T defined by Post a Idec Post a 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with � F = {I | I ∩ F ̸ = � }
Ideal completion Definition ([BFM14,FG12]) 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with � F = {I | I ∩ F ̸ = � } ( ) T defined by Post � � W Post W a ( I ) = Idec a ( I ) ↓
Ideal completion Definition ([BFM14,FG12]) • deterministic deterministic • Lemma 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with � F = {I | I ∩ F ̸ = � } ( ) T defined by Post � � W Post W a ( I ) = Idec a ( I ) ↓ • � W finitely branching
Ideal completion Definition ([BFM14,FG12]) • Lemma 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with � F = {I | I ∩ F ̸ = � } ( ) T defined by Post � � W Post W a ( I ) = Idec a ( I ) ↓ • � W finitely branching ⇒ � • W deterministic = W deterministic
Ideal completion Definition ([BFM14,FG12]) Lemma 26 Let W = ( S , ⩽ , T , I , F ) WSTS Its ideal completion is W = ( {I ⊆ S | I ideal } , ⊆ , � � T , Idec ( I ↓ ) , � F ) with � F = {I | I ∩ F ̸ = � } ( ) T defined by Post � � W Post W a ( I ) = Idec a ( I ) ↓ • � W finitely branching ⇒ � • W deterministic = W deterministic • L ( � W ) = L ( W )
Using the ideal completion Proposition 27 If X is an inductive invariant for W , then its ideal decomposition Idec ( X ) ↓ is a finitely-represented inductive invariant for � W .
Using the ideal completion Proposition Proof. Property of being an inductive invariant carries over 27 If X is an inductive invariant for W , then its ideal decomposition Idec ( X ) ↓ is a finitely-represented inductive invariant for � W . Any set of the shape Idec ( Y ) ↓ is finitely-represented in � W
Using the ideal completion Proposition Proof. Property of being an inductive invariant carries over 27 If X is an inductive invariant for W , then its ideal decomposition Idec ( X ) ↓ is a finitely-represented inductive invariant for � W . Any set of the shape Idec ( Y ) ↓ is finitely-represented in � W Result in particular applies to Cover = Post ∗ ( I 1 × I 2 ) ↓ .
Using the ideal completion Proposition Proof. Property of being an inductive invariant carries over 27 If X is an inductive invariant for W , then its ideal decomposition Idec ( X ) ↓ is a finitely-represented inductive invariant for � W . Any set of the shape Idec ( Y ) ↓ is finitely-represented in � W Result in particular applies to Cover = Post ∗ ( I 1 × I 2 ) ↓ . Remark: � W is not necessarily a WSTS.
Conclusion
Regular separability for WSTS languages Theorem If two WSTS languages are disjoint, 28 one of them finitely branching or deterministic or ω 2 , then they are regularly separable.
Also in the paper... 1. A similar result for downward-compatible WSTS Theorem If two DWSTS languages, one of them deterministic, are disjoint, then they are regularly separable 29
Also in the paper... 1. A similar result for downward-compatible WSTS Theorem If two DWSTS languages, one of them deterministic, are disjoint, then they are regularly separable 2. A size estimation for the case of Petri nets Theorem Given two Petri nets, their coverability languages can be separated by • Upper bound: an NFA of triply-exponential size • Lower bound: a DFA of triply-exponential size 29
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